Calculate L^(-1){(1)/(s^2(s^2+a^2))} using the convolution theorem L{f ** g}=L{f} * L{g}

Janet Hart 2022-09-22 Answered
Calculate L 1 { 1 s 2 ( s 2 + a 2 ) } using the convolution theorem L { f g } = L { f } L { g }
My approach:
f ( t ) = L 1 { 1 s 2 } = t
g ( t ) = L 1 { 1 s 2 + a 2 } = 1 a sin ( a t )
L 1 { 1 s 2 ( s 2 + a 2 ) } = f g = f ( t τ ) g ( τ ) d τ = 1 a ( t τ ) sin ( a τ ) d τ
but the last integral is clearly divergent.
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Answers (1)

Nancy Phillips
Answered 2022-09-23 Author has 12 answers
Note that we have
f ( t ) = L 1 { 1 s 2 } = t u ( t )
and
g ( t ) = L 1 { 1 s 2 + a 2 } = sin ( | a | t ) | a | u ( t )
Then, application of the convolution theorem yields
L 1 { 1 s 2 ( s 2 + a 2 ) } = ( f g ) ( t ) = f ( t τ ) g ( τ ) d τ = ( t τ ) u ( t τ ) sin ( | a | τ ) | a | u ( τ ) d τ = 0 t ( t τ ) sin ( | a | τ ) | a | d τ
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